Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #37528 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,19)(14,17)(15,16)(18,21)(20,24)(22,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,38)(39,40);(1,2,4,6)(3,5,8,7)(10,16)(13,19)(17,23)(21,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,13,16,19)(14,18,22,20)(17,21,23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(3,8)(5,7)(9,12,15,11)(10,13,16,19)(14,18,22,20)(17,24,23,21)(25,26)(27,28)(29,37)(30,33)(31,34)(32,38)(35,40)(36,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,12)(11,16)(14,21)(15,19)(17,20)(18,23)(22,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,14)(10,17)(11,18)(12,20)(13,21)(15,22)(16,23)(19,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315263);